Question 1
probability between 2.8 and 3.3
The graph of the normal distribution is shown in the diagram below. We first need to standardise the value of X=2.8 and value X=3.3. Standardising X is just another word for finding z-score
z-score for X = 2.8
[tex]z= \frac{2.8-3.1}{0.41} =-0.73[/tex] (the negative answer shows the position of X = 2.8 on the left of mean which has z-score of 0)
z-score for X = 3.3
[tex]z= \frac{3.3-3.1}{0.41}=0.49 [/tex]
The probability of the value between z=-0.73 and z=0.49 is given by
P(Z<0.49) - P(Z<-0.73)
P(Z<0.49) = 0.9879
P(Z< -0.73) = 0.2327 (if you only have z-table that read to the left of positive value z, read the value of Z<0.73 then subtract answer from one)
A screenshot of z-table that allows reading of negative value is shown on the second diagram
P(Z<0.49) - P(Z<-0.73) = 0.9879 - 0.2327 = 0.7552 = 75.52%
Question 2
Probability between X=2.11 and X=3.5
z-score for X=2.11
[tex]z= \frac{2.11-3.1}{0.41}=-2.41 [/tex]
z-score for X=3.5
[tex]z= \frac{3.5-3.1}{0.41} =0.98[/tex]
the probability of P(Z<-2.41) < z < P(Z<0.98) is given by
P(Z<0.98) - P(Z<-2.41) = 0.8365 - 0.0080 = 0.8285 = 82.85%
Question 3
Probability less than X=2.96
z-score of X=2.96
[tex]z= \frac{2.96-3.1}{0.41}=-0.34 [/tex]
P(Z<-0.34) = 0.3669 = 36.69%
Question 4
Probability more than X=3.4
[tex]z= \frac{3.4-3.1}{0.41}=0.73 [/tex]
P(Z>0.73) = 1 - P(Z<0.73) = 1-0.7673=0.2327 = 23.27%
